Microsoft PowerPoint - chap0B_EM_Theory []Textbook, Sec. 1.3, 1.6, 1.9, 1.8
What will you learn • Maxwell’s eq.
Maxwell
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Content
• B. Maxwell Equations • C. Poynting Vector y g • D. Solution of Helmholtz Eq. • E. Equivalence Principle • F. Duality & Babinet’s Principle • G. Reciprocity Theorem
National Taiwan UniversityRef. Textbook, Section 1.1, 1.2, 1.3, 1.6, 1.8, 1.9, 1.10
B.1 Time-Domain Eq. in Free Space
μ0B = H∂ ∇× = −
B E
ε0D = E
Electromagnetic Field,” 1865. James Clerk Maxwell (1831-79)
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1 ( , ) Re ( , ) ( , ) ( , ) 2
∞
jω∂ ⇒
j jω μ μ′ ′′∇× = − −E B, B = H, =μ μ
For linear, isotropic media
D = E
μ μ
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∫ EE
0
i
i
t
ρ ∂
r e d
π ∞
−∞ = −∫ ( )E
• Linear isotropic homogenous simple media
= = ; ( ) = real constσ ε μ σ ε μ=J E D E B H, = , = ; ( , , ) = real const.c σ ε μ σ ε μ=J E D E B H
- Time-domain
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J EE H H E E H H E
( ) 2 2
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2 2
− ⋅ = ⋅ + ⋅ + ⋅ + × ⋅ ∫ ∫ ∫ ∫J E EE E E H H E H n
S
= + + +P P PW W Ω
Power supplied by the source Ji
in Ω Power dissipated
Power flow out of Ω through S
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=
0 0
0 0
0 0
t V t V e e e
t I t I e e e
t t t V t I t
θ ω ω
ω ω
1 1
T θ
1 1 < ( ) < ( ) ( ) cos( ) Re Re 2 2
t t t V I θ >= >= − = = VI pp v i
* ( ) 0 0
jV I e θ −=p VI
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ti titi
j t j tr t r e r t r eω ωω ω = = E H E H
< 2
Ω Ω
dS dS P
×p E H
Average power flow
Maxwell eqs. i
( ) ( )
μ ε
S
− ⋅ = ⋅ + ⋅ − ⋅ + × ⋅
∫ ∫ ∫ ∫E J E E H H E E E H
( ) 1 + 2 2i d m eP j W W dSω= − + × ⋅∫ *P E H
n
Ω
S
( ) ( )* * * * * * * - ij jωμ σ ωε∇⋅ × = ⋅∇× ⋅∇× = − ⋅ − ⋅ + −*E H H E E H H H E J E E *
* * * * *1 1 1 2 2 2 4 4 2
i j μ εσ ω − ⋅ = ⋅ + ⋅ − ⋅ +∇ ⋅ ×
E J E E H H E E E H
iJ
2S ∫
( )
Remin source free region, incoming complex power 1 + 2 2 d m e
S
Ω S
0= ∂
+⋅∇ t
ω με ρ ε
0
0
f R r r R R
f r r f
Case (i):
R r V
Case (ii): 1 1 ( ) ( ) ;
∈
= ∇ = ∇ ⋅∇ ∫ ∫
πδ ∇ = − − = −
x0z0
2
S S dS R Rd
R R r rP r dS P r dS P r
R R
ε ε
ε ε
π =− Ω
D.3 Sol. of Inhomogeneous Helmholtz Eq.
( )2 2inhomogeneous Helmholtz eq.: k μ∇ + = −A J 21jkRe k− 2 2
0 1 ; jkR jkRe ke e R r r
R R R − − ∇ = ∇ − = − ⇒
1 1 4 ; jkR
πδ −
μ π
02
′∂ ∂ ∂ ∇ = + = ⋅ =
∂ ∂ ∂
nTx E,H E, H E1,H1
E, H
Tx
Original problem
Equivalent problem 0 0 0 0( ) ( ) ; ( ) ( )
4 4
jkR jkR
μ ε π π
Let ( , ) be an electromagnetic field in a region ;
( , ) be another field in . Consider the difference ( , )
Ω
Ω − −
E H
E H E E H H2 2 1 2 1 2( , ) ( , )
( ) ( )1 2 1 2It is zero if either n or n vanishes along .E E H H S− × × −
lossy: either 0, 0, or 0
i σ ε μ′′ ′′≠ ≠ ≠⇓
ΩE E H H
( ) ( ) ** 2 2 2
1 2 1 2 1 2 1 2 1 2 1 12 2 4 4 2S
dS j H E E dvμω σ Ω Ω
− × − ⋅ = − − − + −
n
Ω
S
1 2 1 2; = in = ΩE E H H
The uniqueness theorem may fail for lossless case at certain discrete ω cases that correspond to resonant modes.
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nE H E,H E=H=0 E,H ˆ
ˆm n= − ×K E
n= − ×mK E
Due to the necessity to match the b d di i h i i l
1=H
PMC n
boundary condition, the principle does not give any computational advantage but to provide an alternative way of formulating a BVP, except …
Schelkunoff’s Equiv. for Plane Surface
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J M
J M J M
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J M J M
Applications of Image Theory
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F.1 Duality in Maxwell’s Eqs. Electric sources, Magnetic sources, J M
Electric & Magnetic Dual Quantities
j
ωμ
ωε
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b c
21 HE κ−= dI
abcda ⋅= ∫ 11 H dI
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Application - CPW Inductance Calculations
• CPW inductance CPS capacitance
National Taiwan UniversityRef.: C. W. Chiu and R. B. Wu, "A moment method analysis for coplanar waveguide
discontinuity inductances," IEEE T-MTT, vol. 41, pp.1511-1514, Sept. 1993
1 4CPW CPS CPW CPSH kE L Cμ ε= ⇒ =
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Consider two completely independent (or unrelated) problems with same frequency and environment (symmetric media),
1 1 1
1 1 1
ωμ ωε
∇× = − − ∇× = +
( ) ( ) ( ) ( ) ( )
1 2 2 1 2 1 1 2 1 2 2 2
1 2 1 2 2 1 2 1symmetric media
E H E H H E E H H E E H
J E M H J E M H
∇⋅ × − × = ⋅∇× − ⋅∇× − ⋅∇× − ⋅∇×
= ⋅ − ⋅ − ⋅ − ⋅
ωμ ωε
∇× = − − ∇× = +
( ) ( ) ( )1 2 2 1 1 2 1 2 2 1 2 1E H E H dS J E M H dV J E M H dV× − × ⋅ = ⋅ − ⋅ − ⋅ − ⋅∫ ∫∫ ∫∫ General form
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( )1 2 1 2 1 2Reaction: ,s f J E M H dV Ω
≡ ⋅ − ⋅∫∫ S Ω Ω
radiation at ∞ ( ) 1 2 2 1
S
Applications • Form A: source free,
( ) 0E H E H dS× − × ⋅ =∫ 1 1 2 2, , , 0 in J M J M = Ω n
S
• Form B: suitable B.C., e.g. – PEC, tangential E field = 0, or PMC, tangentail H field = 0 – Radiated field, S = S∞
( )1 2 2 1 0 S
E H E H dS× × =∫ waveguide (cavity) problems Ω
( ) ( )1 2 1 2 2 1 2 1 2 1 1 2; i.e., , ,J E M H dV J E M H dV f s f s⋅ − ⋅ = ⋅ − ⋅ =∫∫ ∫∫
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– Reaction of field of system 1 on source of system 2 Ω Ω antenna problems
≡ ⋅ − ⋅∫∫
Ex. B. N. Das et al., “Resonant conductance of inclined slots in the narrow wall of a rectangular wavegude,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 759-761, 1984.
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V
Δ −
= JE
Δ −
= JE
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• Antenna gain is the same whether used for receiving or transmitting
Did you Learn -1 • Can you write down Maxwell eq. in time domain?
In frequency domain? • What is physical meaning of Poynting vector? Can
you remember the Poynting theorem in time domain? In frequency domain?
• How potential functions are defined in relevance to E and H field? Can you write down its PDE
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w.r.t. current and charge source? • Can you solve the PDE to obtain E and H?
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Did you Learn -2 • What is Love’s equ. principle? Schelkunoff’s equ.
Principle? The difference. • How image principle is related to equ. Principle?
How to apply it? • Can you write down the duality between E and H? • What is Babinet’s principle? Under what
condition, it can be applied? Give an example?
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condition, it can be applied? Give an example? • What is reciprocity theorem? Can you write down